Displacement sensing over large distances may involve the summation of multiple measurements from multiple steps where each step represents a distance between two points. Since each measurement is typically a stochastic process, measurement errors tend to accumulate in a random walk fashion. Although the error from each step may be small, the individual errors may accumulate. As a result, the accumulated measurement errors may reach unacceptably high levels.
For example, assume that a displacement takes 10,000 measurements and that each measurement error accumulates. Assuming that the measurement error is random and the distribution of a large number of measurements will then follow a random walk distribution (i.e., a simple stochastic process where each measurement has an error with a random magnitude and direction), this displacement provides a final measurement error that is equal to the product of the individual measurement step error and the square root of the number of measurement steps, Nstep. Accordingly, the overall measurement error and the individual measurement errors, estep, may be expressed as standard deviations as shown in Equation I.σfinal=√{square root over (Nstep)}×estep=√{square root over (10,000)}×estep=100×estep  Equation I
It would be desirable to perform displacement sensing over large distances while minimizing the overall measurement error in a displacement.